Exploring Measures of Central Tendency: Mean, Median, and Mode

In informetrics and scientometrics, we constantly deal with numbers: how many times a paper is cited, how many articles an author publishes, how many journals carry a particular subject. But raw lists of numbers tell us very little on their own. To make sense of a dataset, we first need to find its “centre” – a single value that represents the whole group. This is exactly what measures of central tendency do. They condense an entire distribution into one typical value, helping researchers, librarians, and information scientists summarise and compare data quickly. The three classic measures are the mean, the median, and the mode, and each one tells a slightly different story about your data.

What are measures of central tendency? 🔗

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that dataset. Because of this, these measures are sometimes called measures of central location, and they belong to a broader family known as summary statistics. The idea is simple: most data points in a distribution tend to cluster around a middle value, and central tendency captures where that clustering happens.

The three most common measures are the mean, median, and mode. Each calculates the central point using a different method, and choosing the right one depends largely on the type of data you have and the shape of its distribution. They work best in combination because they have complementary strengths and limitations. Sometimes only one or two of them will even apply, depending on whether your data is numerical or categorical.

The mean: the arithmetic average 🔗

The mean, often simply called the average, is the most familiar measure of central tendency. You calculate it by adding up all the values in a dataset and then dividing by the number of values. In statistical notation, the sample mean is written as x̄ (x-bar) and the population mean as μ (mu), though the calculation procedure is identical for both.

Suppose five journals in a library collection received the following number of citations in a year: 12, 15, 18, 20, and 25. The mean is (12 + 15 + 18 + 20 + 25) ÷ 5 = 90 ÷ 5 = 18 citations. This tells us that, on average, each journal attracted about 18 citations.

Why the mean is widely used 🔗

The mean’s biggest strength is that it uses every single value in the dataset. Because of this, it is the most informative measure when data is symmetrically distributed, and it is the building block for many advanced statistical techniques such as standard deviation and variance. For normally distributed data, the mean is widely preferred as the best measure of central tendency precisely because it incorporates all the values.

The limitations of the mean 🔗

The mean has one serious weakness: it is highly sensitive to outliers, those unusually large or small values that sit far from the rest of the data. A single extreme value can pull the mean in its direction and make it unrepresentative of the typical data point.

Consider author productivity, a common topic in scientometrics. Imagine nine researchers in a department who published 2, 2, 3, 3, 4, 4, 5, 5, and 50 papers respectively. That last figure of 50 belongs to one extraordinarily prolific scholar. The mean works out to about 8.7 papers per author – yet eight of the nine researchers published five papers or fewer. Here the mean badly misrepresents the group because of one extreme value. This kind of skew is extremely common in bibliometric data, where a small number of authors or papers account for a disproportionately large share of the output.

The median: the middle value 🔗

The median is the middle number in an ordered dataset. To find it, you first arrange all values in ascending (or descending) order and then locate the central value. If there is an odd number of observations, the median is simply the middle one. If there is an even number, the median is the arithmetic mean of the two middle values.

Returning to the nine researchers from earlier – 2, 2, 3, 3, 4, 4, 5, 5, 50 – the data is already ordered, and the middle (fifth) value is 4. So the median is 4 papers. Notice how much more representative this is of the typical researcher than the mean of 8.7. The prolific outlier barely affects the median at all.

The median’s advantage in skewed distributions 🔗

This robustness is the median’s defining feature. Because it depends only on the position of values rather than their magnitude, it is largely unaffected by extreme outliers or non-symmetric distributions. As a result, the median is usually the preferred measure of central tendency whenever a distribution is skewed, which describes most real-world informetric data.

The Australian Bureau of Statistics notes that the median is less affected by outliers and skewed data than the mean, making it the usual preferred measure when a distribution is not symmetrical. This is why median values are so often reported for things like income, house prices, and citation counts, all of which tend to have long right-hand tails.

Where the median falls short 🔗

The median has its own limitations. Unlike the mean, it is not amenable to further mathematical calculation and is therefore not used in many advanced statistical tests. It also ignores the actual size of most values in the dataset, using only their order. And because it requires data that can be ranked, the median cannot be calculated for purely categorical (nominal) data such as subject categories or document types.

The mode: the most frequent value 🔗

The mode is the value that occurs most frequently in a dataset. To find it, you simply count how often each value appears and identify the one with the highest frequency. A dataset can have no mode at all (if every value occurs only once), one mode, or even multiple modes. When two values share the highest frequency, the distribution is called bimodal.

For example, if the most commonly issued book category in a library over a week was “Fiction” on 40 occasions, compared with fewer issues for other categories, then “Fiction” is the mode.

The mode and categorical data 🔗

The mode has one unique and powerful advantage: it is the only measure of central tendency that can be used with nominal data, that is, categorical data without any natural order. You cannot calculate an average subject or a median document type, but you can certainly identify the most frequent one. This makes the mode indispensable in library and information work, where we frequently summarise categorical variables.

According to the comparison of measurement levels, the mode can be used for any level of measurement but is most meaningful for nominal and ordinal data. In practical terms, the mode answers everyday questions like: which is the most borrowed genre, the most common shoe size, or the most frequently searched keyword. The Australian Bureau of Statistics highlights this versatility, noting the mode can be found for both numerical and categorical data alike.

The limitations of the mode 🔗

The mode is the least used of the three measures for numerical analysis. It is not algebraically defined, so it cannot be used in further statistical calculations, and it can be unstable when the sample size is small, since a small change in the data can shift the most frequent value considerably. The mode can also be misleading when the most common value sits far away from the bulk of the data. For these reasons it is rarely used as a standalone summary statistic for continuous data, except to describe bimodal distributions.

Calculating central tendency from grouped data 🔗

So far we have worked with individual values. But in real research, especially with large datasets, data is often organised into a frequency distribution with class intervals – for example, “0-10 citations,” “11-20 citations,” and so on. When data is grouped this way, we lose the individual values and must use special formulas based on class intervals and frequencies.

Mean of grouped data 🔗

For grouped data, we first find the class mark (the midpoint of each class interval, calculated as the average of its lower and upper limits). The direct-method mean is then the sum of each frequency multiplied by its class mark, divided by the total frequency: x̄ = Σ(fᵢxᵢ) ÷ Σfᵢ. When the numbers are large, statisticians often use the assumed mean method or the step deviation method to simplify the arithmetic. The step deviation formula is x̄ = A + (Σfᵢuᵢ ÷ Σfᵢ) × h, where A is the assumed mean, h is the class width, and uᵢ = (xᵢ − A) ÷ h.

Median of grouped data 🔗

To find the median of grouped data, you first build a cumulative frequency table and locate the median class – the class where the cumulative frequency first reaches N/2. The formula is Median = L + ((N/2 − cf) ÷ f) × h, where L is the lower boundary of the median class, N is the total frequency, cf is the cumulative frequency of the class before the median class, f is the frequency of the median class, and h is the class width. The worked examples for this formula show how each term is read directly off the cumulative frequency table.

Mode of grouped data 🔗

For grouped data, we first identify the modal class, the class interval with the highest frequency. The mode is then estimated using the formula Mode = L + ((f₁ − f₀) ÷ (2f₁ − f₀ − f₂)) × h, where L is the lower boundary of the modal class, f₁ is the frequency of the modal class, f₀ is the frequency of the class before it, f₂ is the frequency of the class after it, and h is the class width.

The empirical relationship and skewness 🔗

The three measures are not independent of one another. The way they line up actually reveals the shape of the distribution. In a perfectly symmetrical distribution, the mean, median, and mode all coincide at the same value. In a positively skewed distribution (a long tail to the right, common in citation data), the order is Mean > Median > Mode, because the mean is pulled toward the high outliers. In a negatively skewed distribution, the order reverses to Mean < Median < Mode.

For a moderately skewed distribution, Karl Pearson observed a useful approximation now called the empirical relationship: Mode = 3 Median − 2 Mean. This lets you estimate any one measure when the other two are known. For instance, if an analysis of monthly salaries gives a median of ₹45,000 and a mean of ₹52,000, the estimated modal salary is 3(45,000) − 2(52,000) = ₹31,000. Keep in mind this is only an approximation and works best for moderately skewed, grouped continuous data, not for highly skewed or discrete datasets.

Choosing the right measure 🔗

There is no single “best” measure of central tendency; the right choice depends on your data. Use the mean for symmetrical numerical data where you need a measure that uses every value and feeds into further analysis. Switch to the median when your data is skewed or contains outliers, as is typical in bibliometrics and income studies. Reach for the mode when working with categorical data or when you specifically want to know the most common category or value. In good practice, researchers report more than one measure together so that readers get a fuller, more honest picture of the distribution.

What do you think? If you were summarising the citation counts of papers in a research database – where a handful of landmark papers attract thousands of citations while most attract only a few – which measure of central tendency would give the most honest picture of “typical” impact, and why? And can you think of a situation in library work where reporting only the mean might actively mislead a decision-maker?

References

1. https://statistics.laerd.com/statistical-guides/measures-central-tendency-mean-mode-median.php
2. https://pmc.ncbi.nlm.nih.gov/articles/PMC3157145/
3. https://www.geeksforgeeks.org/maths/measures-of-central-tendency/
4. https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/measures-central-tendency
5. https://www.scribbr.com/statistics/central-tendency/
6. https://www.geeksforgeeks.org/maths/mean-median-and-mode-of-grouped-data/
7. https://byjus.com/maths/relation-between-mean-median-and-mode/